Factors, Multiples and Primes
Primary 5 · Mathematics
Factors, Multiples and Primes
Lesson 3 of 3
14 min
It is 7:00 in the morning at Nyabugogo bus park. The bus to Musanze leaves every 15 minutes. The bus to Huye leaves every 20 minutes. Right now, both of them are pulling out of the gate together.
Habimana is watching, and he wonders: when will I next see two buses leave at exactly the same moment?
That question — and its twin, "how do I share things into the biggest fair groups?" — is what today's lesson answers.
Write down when each bus leaves, counting minutes after 7:00.
Musanze bus, every 15: 15, 30, 45, 60, 75, 90 …
Huye bus, every 20: 20, 40, 60, 80 …
Those are just the multiples of 15 and the multiples of 20. The first number that appears in both lists is 60. And 60 minutes after 7:00 is 8:00. That is Habimana's answer.
60 is a common multiple of 15 and 20, and it is the smallest one there is. We call it the lowest common multiple, or LCM.
(120 is a common multiple too — the buses meet again at 9:00 — but Habimana asked for the first time, so the LCM is what he needs.)
Try 8 and 12. List the multiples of each until the lists shake hands.
Multiples of 8: 8, 16, 24, 32, 40, 48 …
Multiples of 12: 12, 24, 36, 48 …
The first match is 24. So the LCM of 8 and 12 is 24.
Watch out for the trap. Many pupils just multiply: 8 × 12 = 96. Ninety-six is a common multiple — but it is not the lowest one. Multiplying only gives the right answer when the two numbers share no factors at all (like 5 and 7).
Now a different kind of problem. Keza has 30 mandazi and 45 bananas to hand out at umuganda. Every basket must hold the same number of mandazi and the same number of bananas, with nothing left over — and she wants to make as many baskets as she can.
Think about what the number of baskets must do: it has to divide 30 exactly and divide 45 exactly. So it must be a factor of both. List them.
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 45: 1, 3, 5, 9, 15, 45
The factors sitting in both lists are 1, 3, 5 and 15. The biggest is 15.
So Keza makes 15 baskets. Each one gets 30 ÷ 15 = 2 mandazi and 45 ÷ 15 = 3 bananas. Nothing is left over, and she could not have made even one basket more.
15 is the greatest common factor of 30 and 45 — the GCF.
For big numbers, listing gets slow. Prime factorisation does the job faster. Take 24 and 36.
First break each one into primes (you learned this last lesson):
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
The GCF is what they SHARE. Line them up and look. Both have a 2. Both have a second 2. Both have a 3. But 24 has a third 2 that 36 cannot match, and 36 has a second 3 that 24 cannot match — so those two cannot be shared.
GCF = 2 × 2 × 3 = 12
The LCM is EVERYTHING, taking the most of each prime. The most 2s in either number is three (24 has three). The most 3s is two (36 has two).
LCM = 2 × 2 × 2 × 3 × 3 = 72
Check by listing: multiples of 24 are 24, 48, 72 … multiples of 36 are 36, 72 … It matches. And 12 really does divide both 24 and 36 (giving 2 and 3). ✓
Here is a lovely check you can use on any pair: GCF × LCM = the two numbers multiplied together. Look: 12 × 72 = 864, and 24 × 36 = 864. They agree.
Tip
How do I know which one the question wants? Ask what is happening in the story.
Say it to yourself: meeting up is Multiples; cutting up is Factors.
Decide LCM or GCF first — then work it out.
Find the LCM of 6 and 8.
Answer: multiples of 6 are 6, 12, 18, 24 … multiples of 8 are 8, 16, 24 … The first match is 24. (Not 48 — that is 6 × 8, a common multiple, but not the lowest.)
Find the GCF of 18 and 27.
Answer: factors of 18 are 1, 2, 3, 6, 9, 18. Factors of 27 are 1, 3, 9, 27. The common ones are 1, 3 and 9. The greatest is 9.
Find the LCM of 5 and 7.
Answer: 5, 10, 15, 20, 25, 30, 35 … and 7, 14, 21, 28, 35 … The LCM is 35. Here it really is 5 × 7, because 5 and 7 share no factor except 1.
Mugisha waters his beans every 4 days and his maize every 6 days. He watered both today. In how many days will he water both on the same day again?
Answer: the jobs repeat and meet up, so this is an LCM. Multiples of 4: 4, 8, 12. Multiples of 6: 6, 12. He waters both again in 12 days.
A school has 36 pencils and 48 exercise books to share out equally among as many pupils as possible, with none left over. How many pupils, and what does each pupil get?
Answer: things are being cut into equal groups, so this is a GCF. Use primes: 36 = 2 × 2 × 3 × 3 and 48 = 2 × 2 × 2 × 2 × 3. What they share is 2 × 2 × 3 = 12. So 12 pupils, each getting 36 ÷ 12 = 3 pencils and 48 ÷ 12 = 4 books.
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